Some times ago I posted this question here. There I carelessly assumed that if you have a set of sections of a vector bundle which span every fiber pointwise, they also generate the module of smooth sections over the smooth functions on the base manifold, I think that is true if your set of sections is finite. But in my situation for the horizontal bundle this is not the case. So I want to ask the following question:
Let $(P,\pi,B,G)$ be a principal bundle with total space $P$, base $B$, projection $\pi$ and structure group $G$. Let $\gamma \colon TP \to \mathrm{Lie}(G)$ a connection one-form, which induces the horizontal space $HP$. For a vector field $X \in \Gamma^\infty(TB)$ I write $X^h \in \Gamma^\infty(HP)$ for the horizontal lift of $X$ with respect to $\gamma$.
Is it true that $\Gamma^\infty(HP) = C^\infty(P)-\mathrm{Span}(X^h \mid X \in \Gamma^\infty(TB))$, i.e. that the horizontal lifts generate the $C^\infty(P)$-module of horizontal vector fields.
If so, can you give me a proof?
(For a vector bundle $E \to M$ I denote the smooth sections on $E$ by $\Gamma^\infty(E)$. $C^\infty(M)$ are the smooth functions on the manifold $M$)
Note that every manifold is assumed to be real, finite dimensional, $T_2$, paracompact and smooth.
